This article ischaracterized by approaching oneofthe problems encountered in port systemsknown as the Berth Allocation Problem. As the operational activities require a high degree oftime to be carried out and, in most cases they are done manually, it becomes necessary to usean optimization tool. To obtain a good solution with a small amount of computational effort, aheuristic model, based on concepts of Genetic Algorithms

(GA), is proposed, allowing thisconcept to be learnt. Prepared generically, with some minor adjustments in the data, themethod can be applied to solve the problem in any port, as the ports use a similar managementsystem. Finally, a numerical experiment examines and evaluates the results, noting theireffectiveness in the aid the improvement and refinement of the management system.

The new global organization based on the creation of global markets requires theestablishment of efficient logistic systems that are capable of allow the efficient flow ofproduction to foreign markets. The ports, as a means of transport, should be considered as animportant link in the integration of the domestic and global markets, and their modernizationis one of the main activities to be prepared with the domestic costs reducing plan, as well asincreasing exports.

It is perceived, therefore, that there is a gap still to be explored regarding research andmethods for one of themost

important operational problems found in the port system: theberth allocation problem

(BAP).

One of the first studies on this subject in Brazil waswritten by

Silva and Coelho (2007),they researched how to determine an allocation plan for the vessels that dock at the port

berths, so that each vessel is placed in a berth for a period of time that is required for theloading and unloading of the cargo.

The berthing plan is to combine the best of the best possible berth to serve each vessel inorder to respect the restrictions that are imposed, resulting in a lower berthing cost. Theattention to these aspects makes the berthing preparation plan atthe port an expensiveoperation and

requires the use of an optimization technique to prepare this plan. Therefore,the general purpose of this article is to propose a heuristic computational model to help thedecision making of a container port complex on the best way to provide the best berth-vesselallocation with the aim of reducing operational costs.

Brown et al. (1994, 1997) addressed the problem of berth allocationin marine ports. Theauthors identified the optimal set of vessel-to-berth attributes that maximizes the sum ofbenefits of vessels while in port. The planning of berths in marine ports has some importantdifferences from the planning of berths in commercial ports. According to Imai et al. (2001),in marine ports the change of berths occurs because of the ports own services, in other words,a new vessel arrives and is assigned to a berth where another vessel is already berthed. Thistreatment is unlikely to

occur in commercial ports. Thus, the exchange of berths and otherfactors less relevant to commercial ports are considered in Brown et al. (1994, 1997) makingthe issue inappropriate for commercial ports.

In the studies done by Lim (1998), the problem was

transformed into a restricted versionof the two-dimensional packing problem. Tong et al. (1999) in Dai et al. (2004) solved theproblem of assigning berths by using the optimization approach used by a colony of ants, butthe focus of the workwas to minimize the length of the required pier.

isfromNishimura et al. (2001) where is proposed a model for dynamic BAP that considers the arrivalof vessels after the start of the elaboration of the mooring plan. For such case,GAsareapplied where two different types of representation for chromosomes are utilized

withsatisfactory results.

Despite the fact that there are many ways to resolve the BAP, the nearestliterature to this article’s line of study is the series of articles of Imai et al. (2001) andNishimura et al. (2001), which will be used as basis for this research development.

The articlediscusses applying GAs to the berth allocation problem andis organized asfollows: in section 2 the formulation of the model is shown, in section 3 the procedure forsolving theGA

is given, in section 4 the numerical experiments and the final sectioncontains

the conclusion of the study and suggestions for future research is given.

2Development of the Model

The objective of this study is to minimize the length of stay of vessels in berths/ports toreduce operating costs. To develop a model that comes close to the maximum of the realproblems are considered extremely important factors which have great impact on theoperational costs, such as the cost of the vessel staying in port (Silva and Coelho, 2008).

In practice it is known that a port receives vessels

of various sizes, therefore, differentrates are charged for the stay. For the development of this article the charges for berthing andhandling of loads shall be considered, which were judged to be the greatest impact in formingthe operational cost of dock side berthing.

2.1Restrictions

Amongst the various influential restrictions in decision making for the berthing of avessel, are:time

restriction,draft restriction,

length

restriction,

vessels schedulingrestriction.

2.2Decision Variables

(when and where toallocate eachvessel)

The docking plan must be capable of allocating vessels to berths in the best possible way,i.e., determine 'what' berth should moor each vessel as well as the 'moment' of berthing,analyzing if the waiting time for each vessel, accounting costs, in order to optimize theoperation of the port system.

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2.3Formulation of the problem:

HeuristicAllocationAlgorithm

Initially a simplified algorithm is proposed that is able to consider a list of vessels (eachcontaining information about thearrival

moment, draft, cargo, length and differentiatedtariffs), and a list of berths (release time, interval, depth, length, productivity rates). Then, it isverified whether the restrictions are met, and subsequently allocates a vessel in each givenberth.

To calculate the total allocation cost

(CA), consider:

Subject to:

Where:

CA: is the objective function of the allocation cost;

Di: draft of vesseli;

Ei: vesseli

waiting time;

Dj: depth of berthj;

Aij: vesseli

service time in the berthj;

TAtracj: wharfage duty charged by berthj

to vesseli;

CPi: Cost of vesseli

stopped;

Ci: load of vesseli;

Mchegi

: arrival moment of vesseli;

TMovj: movement duty charged by berthj

to vesseli;

Mlibj: release moment of berthj;

P: number of executed periods of time;

Li: length of vesseli;

xij: binary variable type 0-1, where value 1 is consideredin the event of vesselihas been serviced in berthj

andvalue 0 is assumed in the event of the contrary;

N = {n1, n2, ..., nk}, represents the entirety of vessels;

B = {b1, b2, ..., bm}, represents the entirety of berths.

The function (1) minimizes the total cost of the allocation,which

is the sum of thefollowing plots:

cost of the waiting time andcost of service time,cost of cargo handling

and

rate of useofthe

berth

length, per period.

The equation (2) represents the restrictions in thelength of the berthj

which must be greater than the length of the vesseli, the restriction (3)indicates that the draft of the vesseli

must be less than the depth of the berthj, the restriction(4) indicates the time the vessel berthed and the service of the vesseli

and this should be morethanthe release moment from berthj, i.e. a vessel cannot be berth and serviced in a givenberth before this is released from servicing the previous vessel.

Because of the fact of greatly variable costs in a sequence of allocation to the other, it isnecessary to evaluate the various possibilities for the sequences that may exist. To be

notexhaustive, it is proposed to use a genetic algorithm to perform the analysis of possibleallocation sequences.

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3 Genetic Algorithm for the Problem’s Solution

GAs are like the heuristic methodswhere

the optimality of the answers cannot bedetermined. They work in the principle of evolving a population of trial solutions over manyiterations to adapt them to the fitness landscape expressed in the objective function.

As shown in the previous section, theBAP

is a non-linear programming problem that isdifficult to solve. To facilitate the solution of the process, a heuristic based on GAs isproposed by Silva and Coelho (2008).

3.1Conditions of theProposedHeuristics

In addition to the factors that were carried out, this paper presents a difference to theNishimura et al. (2001)

model.

The authors work with the division of the problem inn

sub-problems (SUBs) of

berthallocation in terms of the time factor, as in

Fig.1.

Fig.1

-

Berthing Schedule (NISHIMURA et al., 2001).

After the discharge of all the berths, the first sub-problem is solved byGA. Inheriting thesolution of the first SUB (i.e. when

all the berths are free again), the next SUB is solved. Theprocess is repeated until all SUBs

have been resolved, in the meantime the final solution canbe affected by intermediate solutions. In theproposedmodel this does not occur, since it isconsidered as a single issue, containing all the vessels for berthing.

Another difference with the work of Nishimura et al. (2001) is the way to characterize achromosome.The cited authors used

a sequence of random berthing of vessels and then

analyzed

the fitness of this allocation. In the proposed study, the genes are analyzed one byone, choosing the best individual mooring to another vessel and then select another vessel forthe best berthing position. So, a sequence will be formed for berthing, in other words achromosome.

3.2GeneticAlgorithm

The GA as a heuristic methoddoes

not guarantee the optimality of the responsesbutsearch for the adequate solution of the problem.

This algorithm can be understood better byfollowing the steps below:

Step 1.

Generate population of individuals.

Step 2.

Calculate value of objective function.

Step 3.

Select ancestors.

Step 4.

Make the crossover between ancestors.

Step 5.

Perform the mutation.

Step 6.

Calculate value of objective function.

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Step 7.

If the value of the new objective function is better than theprevious value of the objective function, go to Step 8.Otherwise, go to Step 9.

In the proposed method,a sequence of vessels, which can carry with themselves thevalues of the variables (depth, length, cargo) is considered as a chromosome:

WhereNnk

represents thekthvessel

from

the list,and eachvessel

has a

position in this list

(for example:vessel

N3 occupies position1in the list).

To sort the individuals of the population during the search process a measurement is usedfor the performance of fitness measured by (1). Being a problem of minimizing the costswhen the lower the value of the objective function, the better the adaptation of the individual,and like this, the method always chooses the best performing individuals causing them toreproduce.

Several rules exist in the literature to make the crossover between individuals andtherefore, for the problem of this research it was decided to differentiate and adopt as a rule,the average of the positions occupied by individuals in the chromosomes(Silva and Coelho,2008).

Consider two chromosomes:

This is the average of the positions occupied by each individual in the two chromosomes:

n1: occupy´s the positions 4 and 3, so the average is 3.5;

n2: occupy´s the positions 2 and 1, so the average is 1.5;

n3: occupy´s the positions 1 and 4, so the average is 2.5

n4: occupy´s the positions 3 and 2, so the average is 2.5;

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Then, list them in ascending order, and when there is tie, randomly picks one of them tofill the position:

As there was tie between

the average of the individuals

n3

andn4,it´schosen

to insert inthe list, in the second position, the vesseln3

andfollowing the vessel

n4.

In this paper was notcompared this method with others encountered in the literature.

After doing the crossover, the mutation takes space.

4 Numeric Validation

4.1

Parameters for the search of the solution

To implement the proposed algorithm to solve the PAB, a software was developed inDelphi® version 7, which was tested and the variables analyzed.

To use the

software

it isneeded toinsert the files that containinformation on thevessels and theberths

(length,draft,duties,arrivalmoment, etc.)and

fill otherinformations like

the

occupation period as well asthe mutation rate to be

considered.Three different ways can be

used as stop criterion:maximumnumber ofiterations achieved;

maximum time of computing achieved and,value of

epsilon1

achieved.

It was considered

three options of resolution methods to be chosen:

Method 1 (based on the earliest release date of thevessels),Method

2

(based on lowest cost ofvessel allocation) andMethod3

(based on the relationship between Method 2 and the lengthof time the vessels remains in theport).

Method 1 starts with a value for the Earliest Departure Date of the vessels as beinginfinite. Next, determine the value of Departure Date that each vessel would have, if the berthand the berthing time varied. The Departure Date is found by the maximum value between theberth release time plus the considered occupation time and the time of arrival of the vesselplus its service time. If the value found for the Departure Date is less than the value for theEarliest Departure Date, this should be replaced by the first one and the procedure berepeated. Subsequently, calculate the value of the objective function using equation (1).

Method 2initiallyaccepts

a value for the minimum cost of allocation as being infinite.

Then, check the best cost of allocation in each of the berths for all vessels, using equation (1).If the value of the Cost is less than the value of the Minimum Cost, it should be replaced bythe first one.

Method 3is similar to the previous method, but differentiates itself from themoment that the value of the cost is found and multiplied by the total time that the vesselstays in the port (from arrival to departure). What is

realized is

that this method potentiates thetime by trying to find the balance between the length of stay of a vessel in the port and costallocation. Because this study has a heuristic nature this method was agreed to analyze thebehavior.

4.2Simulated and real experiments

The proposed method was applied and analysed for some simulated cases.In every case itwas considered as comparing criterion,

a population of3.000 individualsand a mutation rate

1Epsilon

is the adopted parameter to evaluate if the devition obtained between the fitnessof the

worst and thebest

cromossome is aceptable.The deviation is found by the following:

Deviation = (Worst Cromo Fitness–

Best Cromo Fitness) / Best Cromo Fitness

In this manner,ifDeviation > Epsilon,the

algorithm

continues.

7

of

2%,with an occupation

interval

of

2 hours, an occupation period of6 hours

and amaximumnumber ofiterations

of

500.000.

The tests were carried out for the fixed processingtimes of

3, 5 and 10 minutes, also varying the 3 different methods, this totaled 540 examinedcases.

In order to evaluate the proposed system in terms of quality of results, the method wasapplied to the case ofItajaí

Port (Brazil), which currently does not have a tool for theoptimization of allocation of the transaction and, uses Excel® software.

The followinginformation was considered for this test:

each dayfrom November 20 to December 31, 2007was analyzed where

63 vessels were scheduled to berth at the port. Currently the port operates24 hours a day,

7 days a week, with 3 berths.

Itis worth mentioning that

Itajaí

Port has a 740 meter long pier and does not operate usingfixed-size berths. In average there would be 3 berths x 246 meters each, but in practice thislength varies with the length of the vessel that is berthed. For example, if the port receives avessel that is 260 meters long, it will use the first 246 meters of the first berth and another 14meters belonging to the second berth.

To test the developed model

with actual data fromItajaí

were

discarded

vessels with thegreatest length, because the software works with individual berths.

Whereas, the arrival ofthese longer vessels is not very likely, the result of the mooring plan will not be very differentto the plan drawn up by the employees of the port in question.

4.3Results obtained

Due to the number of simulations carried out some tests were prioritized with respect tothe performance obtained by the proposed tool:analysis of the number ofiterationsversus

epsilon

variation;

analysis of the number of iterationsversusthemaximum

processing time;

analysis of the computation time differenceversusthe problemcomplexity and

analysis of thepercentage ofdifferences of values

of the objective function.

With regard to the number of iterations versus

theepsilon

variation, the

three methodshave presented

a decreasing curve, in other words, as the epsilon value increases, the numberof iterations is considerably reduced.

In relation to the analysis of the number of iterationsversus the maximum processing time required, it

was found that during 75.50% of tests, therewas divergence in the number of iterations between methods 1 and 2, i.e. method 1 requiredmore iterations than Method 2 compared to the same period of time. This behavior is alsorepeated when compared to methods 1 and 3.

In this analysis it can also be seen the extent to which there is variation in the number ofvessels for the same amount of berths (e.g. 15 berths, servicing 10, 70 and 100 vessels),increases the difference in the number of iterations between methods 1 and 2 and methods 1and 3.It is notorious that thedifference between

the number of iterations carried out by fixed time ofanalysis.As the complexity of the problem increases,the difference of behaviour between themethods also increases.

With the time convergence analysis of the results, it´s possible do see that by increasingthe complexity of the problem, all three methods tend to generate the optimal solution usingthe same computational time.SeeFig.2.

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Fig.2

-

Difference in the convergence time of between methods 1 and 2.

The values obtained in the objective function (cost allocation) which in most cases wasthe difference between methods 2 and 1 take on the behavior of a growing curve, i.e., as thenumber of vessels increases, it increases the value the percentage difference between themethods, i.e. the value of the difference from method 2 to method 1 grows considerably as thenumber of vessels increases.Fig.3

shows this reasoning:

Fig.3

-

Difference in the best solution found with the methods 1 and 2.

In this case, when the problem has 100 vessels, the method 2 provides resultsin

theobjective function that are 5.61% greater than method 1. As this research attempts to obtain areduction in costs, the greater the value of the solution, the poorer result is.

It should be mentioned that tests carried out in the case ofItajaí

Port were satisfactory.

Itwas possible to allocate all planned vessels and in some cases

there was disagreement as tothe length of stay of vessels in port (in real cases), probably due to exogenous variables (suchas a delay in arrival), which were not considered in the proposed software.

Thus, an analysis was made of the difference in allocation obtained fromItajaí

Port andproposed by the PAB, as shown inFig.4.

Fig.4

-

Analysis of the difference in allocation at the Port of Itajaí and the PAB

The graph contains

the 30 cases which

were analyzed and the difference obtained betweenthe real berthing time at the port and the time proposed by the PAB.It´s possible tosee thatfor most of the cases that were examined, a gain of up to 5 hours was obtained before the

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allocation proposedby the PAB, in other words, the PAB provided the berthing details by upto 5 hours in most of dockings.

The Histogram 1 below shows an easier way tointerpret the allocations made. In

approximately 70% of caseswasshowed a difference in allocation,

between real data and theproposed method,

of up to 5 hours;which shows that the developed model is valid and canprovide satisfactory results.

Histog.

1-

Frequency difference of the allocation in the Port of Itajaí and the PAB

This does not guarantee the efficiency of the proposed method compared to the systeminItajaí Port. For the cases tested in the PAB, it was considered that the berths were madeavailable at the completion of the berthing plan, which in practice, in Itajaí, there wereprobably other vessels occupying the berths and delaying the berthing of vessels used in thisanalysis. However, the importance of the proposed software cannot be discarded, as thesuggested dockings were similar to those that occurred in practice, achieving the proposedgoal of reducing costs as well as the ease of preparing of the berthing plan.

5Conclusions and Recommendations

A heuristic tool to support the decision-making on the best allocation of vessel-berths,usingGAs, was presented in this paper.Although this study has been conducted on otherreferences,it was not possible to comparewith the proposed model, since theanalyzeddatawere not the same.

This method has shown to be effective and with a low degree ofimplementation difficulty, and may be validated for use in container terminals.

As the behavior of simulations varies with the value ofepsilon, it can be confirmed thatthe results were not very stable but, in most cases, the number of iterations was reduced as thevalue ofepsilon

increased.

For the number of iterations and maximum processing time, it was concluded that method1 is more suited to carry out the processing, since it performs more iterations than methods 2and 3. In relation to the results obtained for the valueof the objective function of the threemethods

was

expected

for

method 2 the best result for the allocation

cost, because this was itspurpose: allocateminimizing costs, but it was seen that method 2 presented a greaterallocation cost than method 1 thatincreased the complexity of the problem.

The

decisionof the best methoddepends on the aimed result:lessermooring

time of thevessels, or lesserallocation costs.About the

analysis of convergencetimeof the results it isconcluded that by increasing the complexity of the problem, the methods tend to have similarbehavior reducing the difference between the computational time and results for the optimalsolution.

The proposed tool can have wide applicationsin

the container port system management,facilitating the work of logistics operators,taht

conduct, in most cases, manually the berthingplan. It also promotes the reduction of errors during this activity, as this is

computerized,allowing for a reduction in operating costs.

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To proceed with this research it is recommended to consider the existence of differentcargoes moving around the port (bulk, liquid, etc.), the availability of equipment atthe berths

to receive a certain vessel for handling the cargo, and distance to be traveled by the cargofrom

the vessel to the warehouse or even the method of transport that will be used later. It isalso recommended to change the proposed algorithm to allow more than one vesselberthingto optimize the available space, making a greater number of practical tests